Akkuş, H.; Özkan, E.

Abstract: In this study, we define the -Fermat and -Fermat–Lucas sequences. For these sequences, we give many features such as the characteristic equation and Binet formulas. We obtain the relations between these sequences. Then we find different representations of finding terms of these sequences. Also, we show the relationship between the positive and negative terms of the sequences and get the relationship between three consecutive terms. We present the generating functions and sum formulas of these sequences. In addition, we calculate special identities of these sequences such as Catalan identity, Melham’s identity, d’Ocagne identity, etc. We examine the relationships of -Fermat sequence with Fibonacci, Pell, Jacobsthal, Balancing, Oresme sequences and -Fermat–Lucas sequence with Lucas, Pell–Lucas, Jacobsthal–Lucas, Balancing–Lucas, Oresme–Lucas sequences. Moreover, we present on the application of -Fermat and -Fermat–Lucas sequences to hyperbolic quaternions. Furthermore, we define hyperbolic -Fermat and -Fermat–Lucas quaternions. For these hyperbolic quaternions, we give many properties such as Binet formulas. Finally, the terms of the -Fermat and -Fermat–Lucas sequences are associated with their hyperbolic quaternion values.